Semiparametric estimation of the memory parameter is studied in models of fractional integration in the nonstationary case, and some new representation theory for the discrete Fourier transform of a fractional process is used to assist in the analysis. A limit theory is developed for an estimator of the memory parameter that covers a range of values of d commonly encountered in applied work with economic data. The new estimator is called the modiﬁed local Whittle estimator and employs a version of the Whittle likelihood based on frequencies adjacent to the origin and modiﬁed to take into account the form of the data generating mechanism in the frequency domain. The modiﬁed local Whittle estimator is shown to be consistent for 0 < d < 2 and is asymptotically normally distributed with variance 1/4 for 1/2 < d < 7/4. The approach allows for likelihood-based inference about d in a context that includes nonstationary data, is agnostic about short memory components and permits the construction of valid conﬁdence regions for d that extend into the nonstationary region.
Shimotsu, Katsumi and Phillips, Peter C.B., "Modified Local Whittle Estimation of the Memory Parameter in the Nonstationary Case" (2000). Cowles Foundation Discussion Papers. 1516.